A296839
Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).
Original entry on oeis.org
0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453
Offset: 0
tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...
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nmax = 16; Table[(CoefficientList[Series[Tan[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A296836
Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).
Original entry on oeis.org
1, 1, 2, 3, -3, 20, 105, -5271, 133826, -2714517, 25525845, 2131781300, -235250824479, 17527695547713, -1124258412169438, 58383380825728035, -975024061456732035, -398903577787777972396, 97649546210035758250281, -17069419358223320552890167
Offset: 0
exp(x*tanh(x/2)) = 1 + x^2/2! + 2*x^4/4! + 3*x^6/6! - 3*x^8/8! + ...
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nmax = 19; Table[(CoefficientList[Series[Exp[x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A305711
Expansion of e.g.f. exp(2*x/(exp(x) + 1)).
Original entry on oeis.org
1, 1, 0, -2, -1, 11, 13, -111, -220, 1756, 5051, -39775, -153191, 1215345, 5952668, -48020714, -288569149, 2377190003, 17069110381, -143857868895, -1209439895944, 10435153277620, 101078662547567, -892827447251575, -9834570608359487, 88900938146195601, 1101567283699652888
Offset: 0
exp(2*x/(exp(x) + 1)) = 1 + x - 2*x^3/3! - x^4/4! + 11*x^5/5! + 13*x^6/6! - 111*x^7/7! - 220*x^8/8! + ...
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a:=series(exp(2*x/(exp(x)+1)),x=0,27): seq(n!*coeff(a,x,n),n=0..26); # Paolo P. Lava, Mar 26 2019
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nmax = 26; CoefficientList[Series[Exp[2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k EulerE[k - 1, 0] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
a[n_] := a[n] = Sum[2 (1 - 2^k) BernoulliB[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
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